integral of (x^2+x+1)/(x^3+x)

Lösung

\int \frac{x ^{2} + x + 1}{x ^{3} + x} d x

arctan (x) + ln ∣ x ∣ + C

Schritte zur Lösung

\int \frac{x ^{2} + x + 1}{x ^{3} + x} d x

Multipliziere aus

\frac{x ^{2} + x + 1}{x ^{3} + x} : \frac{x ^{2}}{x ^{3} + x} + \frac{x}{x ^{3} + x} + \frac{1}{x ^{3} + x}

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{2}}{x ^{3} + x} d x + \int \frac{x}{x ^{3} + x} d x + \int \frac{1}{x ^{3} + x} d x

\int \frac{x ^{2}}{x ^{3} + x} d x = \frac{1}{2} ln x^{2} + 1

\int \frac{x}{x ^{3} + x} d x = arctan (x)

\int \frac{1}{x ^{3} + x} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

= \frac{1}{2} ln x^{2} + 1 + arctan (x) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

Vereinfache

\frac{1}{2} ln x^{2} + 1 + arctan (x) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 : arctan (x) + ln ∣ x ∣

= arctan (x) + ln ∣ x ∣

Füge eine Konstante zur Lösung hinzu

= arctan (x) + ln ∣ x ∣ + C

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